Given Irrational Numbers a,b, Is a^b Rational?

Date: 09/26/2001 at 13:53:30
From: JM
Subject: Irrational numbers
Is it possible to demonstrate that there are irrational numbers a, b,
where, when calculating a^b, the result is a rational number?
I've proved it is not impossible because, for example, e^(lna) = a,
and if a*Q then lna*/Q. But I think there must be a systematic
demonstration.
Thank you.

Date: 09/26/2001 at 14:09:50
From: Doctor Paul
Subject: Re: Irrational numbers
Consider the following:
Claim: There exists an irrational number r such that r^sqrt(2) is
rational.
Proof:
If sqrt(3)^sqrt(2) is rational, then we're done because r = sqrt(3) is
the desired value of r.
Otherwise, x = sqrt(3)^sqrt(2) is irrational.
Then x^sqrt(2) = sqrt(3)^2 = 3, which is rational.
Therefore, either sqrt(3) or sqrt(3)^sqrt(2) is an irrational number r
such that r^sqrt(2) is rational.
I think this establishes the result you want - there exist irrational
numbers x and y such that x^y is rational.
- Doctor Paul, The Math Forum
http://mathforum.org/dr.math/